Bottleneck analysis in multiclass closed queueing … · Queueing Systems 31 (1999) 217–237 217 Bottleneck analysis in multiclass closed queueing networks and its application Arthur

Queueing Systems 31 (1999) 217–237 217

Bottleneck analysis in multiclass closed queueingnetworks and its application

Arthur Berger a, Lev Bregman b and Yaakov Kogan c

a Bell Labs, Lucent Technologies, Holmdel, NJ 07733, USAE-mail: [email protected]

b Institute of Industrial Mathematics, Beer-Sheva, IsraelE-mail: [email protected]

c AT&T Labs, Middletown, NJ 07748, USAE-mail: [email protected]

Received 19 March 1998; revised 3 October 1998

Asymptotic behavior of queues is studied for large closed multi-class queueing networksconsisting of one infinite server station withK classes andM processor sharing (PS) stations.A simple numerical procedure is derived that allows us to identify all bottleneck PS stations.The bottleneck station is defined asymptotically as the station where the number of customersgrows proportionally to the total number of customers in the network, as the latter increasessimultaneously with service rates at PS stations. For the case when K = M = 2, the setof network parameters is identified that corresponds to each of the three possible types ofbehavior in heavy traffic: both PS stations are bottlenecks, only one PS station is a bottleneck,and a group of two PS stations is a bottleneck while neither PS station forms a bottleneckby itself. In the last case both PS stations are equally loaded by each customer class andtheir individual queue lengths, normalized by the large parameter, converge to uniformlydistributed random variables. These results are directly generalized for arbitrary K = M .Generalizations for K 6= M are also indicated. The case of two bottlenecks is illustratedby its application to the problem of dimensioning bandwidth for different data sources inpacket-switched communication networks. An engineering rule is provided for determiningthe link rates such that a service objective on a per-class throughput is satisfied.

Keywords: closed queueing networks, asymptotic analysis, bottleneck

1. Introduction

This paper is motivated by a new application of closed queueing networks (CQN)with a large number of customers. The application is the dimensioning of bandwidthfor different data sources subject to feedback control in packet-switched communica-tion networks when available bandwidth at the servers is shared between all activesources. In a CQN, data sources are modeled by an infinite server (IS) station, andnetwork nodes are modeled by processor sharing (PS) stations. It is known that thesteady state queue length distribution in such a CQN has a product form that is defined

J.C. Baltzer AG, Science Publishers

218 A. Berger et al. / Bottleneck analysis

explicitly up to the normalization constant. The distinguishing property of the newapplication is that this CQN model is adequate only if one or more PS stations forma bottleneck. The bottleneck station is defined asymptotically as the station where thenumber of customers grows proportionally to the total number of customers in thenetwork, as the latter increases simultaneously with service rates at PS stations. Itis known [7] that for a single class CQN (K = 1) consisting of IS and PS stations,in general, only one PS station may be a bottleneck, and the bottleneck node canbe easily identified from the network parameters. Moreover, the asymptotics for themean queue length at the bottleneck station are found from a linear equation. For amulticlass CQN the bottleneck analysis becomes more complicated. For an arbitrarynumber of classes, the bottleneck analysis has been done only in the case when thenumber of PS stations M = 1 [9]. In this case, the asymptotics of the mean queuelength at the bottleneck node are explicitly expressed through the least positive root ofa polynomial of order K, where K is the number of classes. The relative simplicityof the results for K = 1, M > 1 or K > 1, M = 1 is explained by their derivationfrom asymptotic expansions of one-dimensional integral representations for the parti-tion function (normalization constant) in complex [2,6,7] or real [9] space. In general,the integral representations in complex and real space are K- and M -dimensional,respectively, and their asymptotics can be relatively easily derived only in the case ofnormal traffic [7,9] when neither a PS station nor or a group of PS stations forms a bot-tleneck. The bottleneck case requires residue analysis of a K-dimensional generatingpartition function and application of the saddle-point method, which is far from trivialeven in the 2-dimensional case [8], or nontraditional application of the M -dimensionalLaplace method that, to our knowledge, has not been pursued. Moreover, in all casesbut one, the bottleneck conditions given in [8] are quite complicated, their probabilisticinterpretation and generalization for K, M > 2 are unclear, and the case of equallyloaded PS stations is not covered.

Therefore, in this paper, we take a direct approach based on the asymptoticrepresentation for the steady state queue length distribution π(n) derived by Pittel [11].This representation has the following form:

π(n) ∼ C expNF (x)

,

where N is a large parameter (e.g., the total number of customers) and x = n/N .Pittel showed that bottleneck nodes in a large product-form CQN can be identifiedby a nonzero maximum point of some multidimensional function F (x) under naturalconstraints. Positive components of the maximum point x∗ are the limiting values ofqueue lengths, normalized by N , at the bottleneck nodes. Pittel derived these resultsunder the condition that the optimization problem has a unique solution. Not addressedwere the questions: how is this condition expressed in terms of the network parameters,what is the range of network parameters for which the maximum point is not zero,and how to solve the optimization problem.

In our case, the (K ×M )-dimensional function F is found explicitly. We showthat all possible maxima of F under naturally defined constraints can be efficiently

A. Berger et al. / Bottleneck analysis 219

found by the classical Lagrange multiplier method. Using this method we show thatmaximization of F can be reduced to that for an M -dimensional function. Moreover,for this M -dimensional function we derive a simple formula for its partial derivativesthat plays a pivotal role in the bottleneck identification depending on the network para-meters. Finally, the calculation of x∗ is reduced to the solution of algebraic equationsand verification of inequalities. The number of equations and inequalities equals thenumber of bottleneck and non-bottleneck nodes, respectively. In general, these equa-tions are nonlinear. But in an important case when K 6 M and K bottlenecks, x∗

can be found by solving two systems of linear equations of order K.Note that efficient calculation of x∗ is also important for the computation of

the normalization constant in the initial product-form solution for a large networkwith bottlenecks. This is because by its construction the function F (x) is a quasi-potential [4] that provides the logarithmic asymptotic for the product-form solution,and hence expNF (x∗) can be used as a scaling factor in the computation of thenormalization constant.

The outline of the paper is as follows. In section 2 we describe the closedqueueing network, define the scaling under which we study the asymptotics of thesteady state distribution and provide the expression for the function F . In section 3we formulate the main results in two theorems. The first theorem addresses a specialcase of the normalized queue-length limit behavior which is referred to as oscillation.The second theorem provides bottleneck classification for all possible combinationsof the network parameters. We consider the case K = M = 2, but whenever it ispossible the results are formulated in a general form. In section 4 we illustrate thecase of two bottlenecks by its application to the problem of dimensioning bandwidthfor elastic data sources in packet-switched communication networks. In section 5 wefirst establish in three lemmas important properties of the maximum of the function Fand then use them to prove the theorems. In section 6 we indicate generalizations forarbitrary K and M .

2. Asymptotic representation for the steady state distribution

We consider a closed queueing network with K classes and M+1 service stations,one of which is infinite server (IS) and M others are processor sharing (PS) stations.We assume that customers of each class visit all stations. It is convenient to numberthe IS station by 0. Let n denote a K ×M matrix whose element nki represents thenumber of class k customers at PS station i. The population of jobs in class k is aconstant Nk, 1 6 k 6 K. The state space is the set S of matrices n which haveinteger components, and satisfy the population constraints

S =

n | 0 6 nki,

∑i

nki 6 Nk, 1 6 k 6 K, 1 6 i 6M.


Then the product form solution has the form

π(n) =1Ω

K∏k=1

Nk!(Nk −

∑i nki)!

M∏i=1

ni!rnkiki

nki!, (1)

where Ω is the normalization constant and ni =∑

k nki. Moreover,

rki =ekiλkµki

, (2)

where eki is the relative visiting rate of class k jobs to PS station i as compared tothe IS station, 1/λk is the mean service time of a class k job at the IS station, 1/µkiis the mean service time of an isolated class k job at PS station i.

Denote by Qki the random variable for the number of k-type customers in service(queue length) at PS station i and by Q the K ×M matrix of these queue lengths.Random matrix Q takes values n ∈ S. Our goal is to study the limit behavior of Qunder the following assumption:

αk = Nk/N , ρki = Nrki, (3)

where αk and ρki are positive constants while N → ∞. (Note that this scaling isreasonable for the intended application to data networks, section 4, where large valuesof N correspond to the important case of a large number of established connections (orsessions) and to high-speed transmission facilities, i.e., large values of µki and smallrki.) With this assumption we have the following asymptotic representation [11]:

π(n) = C(N ) expNF (x) + O(lnN )

(4)

with

F (x) =∑i

(xi lnxi − xi) +∑k,i

xki ln ρki −∑k,i

xki lnxki

−∑k

(αk −

∑i

xki

)ln

(αk −

∑i

xki

), (5)

where xki = nki/N , xi =∑

k xki for 1 6 k 6 K, 1 6 i 6 M , and C(N ) does notdepend on x. From the definition of the variables xki it follows that x ∈ C, where

C =

x: xki > 0 and

∑i

xki 6 αk. (6)

3. Asymptotic behavior of queues at PS stations

In this section we state the asymptotic results in two theorems and briefly com-ment on them, deferring the proofs to section 5. We consider the case K = M = 2but whenever it is possible the results are formulated in a general form. In section 6we discuss generalizations of the results for different cases when K, M > 2.


Pittel [11] assumed that the function F (x) has a unique maximum point x∗ andproved the convergence in probability of the normalized queue length matrix Q/N tox∗ as N →∞. The first theorem characterizes the limit behavior of Q/N in a specialcase of network parameters where it does not converge to a deterministic limit.

Theorem 1 (Oscillation). If ρki ≡ ρk and∑

k ρkαk > 1 then

1N

∑k,i

QkiP→ v∗ (7)

as N →∞, where v∗ is the unique solution of equation∑k

ρkαkρkv + 1

= 1 (8)

in the interval (0,∑

k αk). Moreover,

Q11 +Q12

NP→ u∗, (9)

where

u∗ = α1 −α1

ρ1v∗ + 1, (10)

andQ11

ND→ U

(u∗), (11)

whileQ21

ND→ U

(z∗), (12)

as N → ∞, where U (d) denotes a random variable with the uniform distribution on[0, d], and z∗ = v∗ − u∗.

The type of the queue length limit behavior at an individual PS station describedby (11) or (12) is referred to as oscillation.

In section 5 we show that the function F (x) has a unique maximum point x∗

except for the special case of network parameters in theorem 1. Moreover,

x∗i =∑k

x∗ki > 0

if and only if x∗ki > 0 for each class k. This property in combination with the

convergence Q/N P→ x∗, [11], justifies the following definition.A PS station i is referred to as bottleneck (non-bottleneck) if x∗i > 0 (x∗i = 0),

given that the maximum point x∗ is unique. Statement (7) addresses a special case,where the bottleneck condition is satisfied only for a group of stations. We say that a


group B = (i1, . . . , im) of m 6M PS stations forms a bottleneck if they are equallyloaded, i.e., ρki ≡ ρk(B), i ∈ B, and the following normalized sum of queue lengths

1N

∑i∈B

∑k

Qki

converges to a deterministic limit v∗(B) > 0 in probability as N →∞.We say that a PS station (a group of equally loaded PS stations) is heavy loaded

if it forms a bottleneck.Representation (4) implies that distribution (1) is concentrated around maximum

points of the function F (x) in the domain C. It turns out that the function F does nothave the unique maximum only in the case when PS stations are equally heavy loadedby each class of customers. This results in random “oscillation” of the individual queuelengths, normalized by N , in contrast to their stabilization to deterministic limits inthe cases of the unique maximum. Load balancing is a plausible design decision inmany applications. However, except for a simple cyclic network consisting of identicalFCFS single servers [5], it was not clear before that although load balancing indeedequalizes the mean queue lengths at different nodes the actual normalized queue lengthsare uniformly distributed random variables.

Denote by ∆ = ρ11ρ22−ρ12ρ21 the determinant of matrix ‖ρki‖. The next theoremconsiders all possible combinations of parameters ρki and αk and provides a completebottleneck classification in the case K = M = 2.

Theorem 2 (Bottleneck classification).1. If

∑k ρkiαk > 1 for all i, ∆ 6= 0 and the two following systems:∑

k

ρkiβk = 1, i = 1, 2, (13)

∑i

ρkiγi = (αk − βk)/βk , k = 1, 2, (14)

have positive solutions, then

x∗ki = ρkiβkγi, (15)

and all PS stations are bottlenecks.2. If

∑k ρkiαk > 1 for i = 1, 2, ∆ = 0 and equations (13) have a solution

βk ∈ (0,αk), then ρki ≡ ρk and the group of two PS stations forms a bottleneck withoscillation at the individual PS stations.

3. There is only one bottleneck PS station in the two following cases:

(i)∑

k ρkiαk > 1 only for one i;

(ii)∑

k ρkiαk > 1 for i = 1, 2 but equation (13) does not have a solution βk ∈ (0,αk),or ∆ 6= 0 and (14) does not have a positive solution.


In case (i) PS station i is the bottleneck. In case (ii) PS station i is the bottleneck if∑k

ρkjαkρkiγ

∗i + 1

< 1 (16)

for j 6= i, where γ∗i is the unique positive solution of equation∑k

ρkiαkρkiγ + 1

= 1. (17)

If PS station i is the only bottleneck, then

x∗ki =ρkiαk

ρki + 1/γ∗i. (18)

4. If∑

k ρkiαk 6 1 for all i, then x∗ = 0, and all PS station are non-bottleneck.

Condition (16) has the following interpretation. By (18) αk−x∗ki = αk/(ρkiγ∗i +1),and the left hand side of the inequality (16) equals

∑k ρkj(αk−x∗ki), which coincides

with the sum of traffic intensities in an open system in which PS station j serves twotypes of Poissonian arrivals whose rates are ekjλkN (αk − x∗ki), k = 1, 2. Thus (16)means that the sum of traffic intensities is less than 1 at a non-bottleneck PS station,i.e., the open system is stable.

We use the classical Lagrange multiplier method to find all possible local maximaof the function F (x) in the domain C. Using this method we derive in section 5 thenecessary conditions for a maximum and reduce the initial problem to maximizationof a function of M variables x1, . . . ,xM . It turns out that the latter function dependsonly on

∑i∈B xi if a group of nodes B forms a bottleneck.

4. Bandwidth dimensioning for elastic data sources

In this section we illustrate statement 1 of theorem 2 by its application to theproblem of dimensioning bandwidth for different data sources in packet-switched com-munication networks, such as Internet Protocol (IP) or Asynchronous Transfer Mode(ATM) networks. For further details see [1]. In our application, a type-k job is a filewith a mean size of fk bits, and the link rate at server i is Li bits per second (bps).Then the service rate of a type-k job at node i (given no other job is present) is

µki =Lifk. (19)

Suppose, for simplicity, that there are two job types generated by finite sources andtwo bottleneck links. This mirrors the important case in data networks under heavyload, where a routing algorithm such as Private Network–Network Interface (P-NNI),[12] directs the traffic to otherwise lightly loaded paths. We model finite data sources


and network nodes by IS and PS stations, respectively, and obtain a CQN model withK = M = 2. Let, also for simplicity,

e11 = p, e12 = 1− p, e21 = q, e22 = 1− q, 0 6 p, q 6 1. (20)

We assume that for a given set of parameters Nk,λk, fk, p, q the link rates Li aresuch that the conditions of statement 1 in theorem 2 are satisfied, i.e., both PS stationsare bottlenecks. Thus we can approximate the sum of the throughputs of type-k jobsat two nodes by

Tk =µk1x

∗k1

x∗11 + x∗21+

µk2x∗k2

x∗12 + x∗22, (21)

where x∗ki are given by (15). Substituting into (21) the values of µki and x∗ki from (19)and (15), respectively, we get

Tkfk =L1ρk1βk

ρ11β1 + ρ21β2+

L2ρk2βkρ12β1 + ρ22β2

= βk(L1ρk1 + L2ρk2), (22)

where the last equality is implied by (13). If we sum the per-class throughputs,k = 1, 2, in (22) and again apply (13) we obtain

T1f1 + T2f2 = L1 + L2. (23)

(23) is the intuitively clear statement that when both PS stations are bottlenecks, thetotal throughput (the sum of the per-class throughputs in bps) equals the total capacity.Thus, given the conditions of statement 1 in theorem 2, from the viewpoint of totalthroughput it does not matter what are the individual values of the link capacities, onlytheir sum. Likewise, note that (23) does not depend on the particulars of the routing,the ekis, other than that statement 1 of theorem 2 pertains. This prediction from themodel matches the rule of thumb in data networks that for a given deployed capacity,the traffic will find the spare bandwidth via the adaptive routing.

Suppose a network designer wants to dimension the capacity of links i = 1, 2to provide a service objective based on throughput. If the chosen objective is interms of the total throughput for both classes, then from (23) the total bandwidth,L1 + L2, simply needs to be equal to the objective on total throughput. If, however,the network designer wishes to offer an objective on per-class throughput, say thebandwidth provided to type-k jobs should be at least Mk bps, then L1 and L2 need tobe chosen such that

Tkfk >Mk, k = 1, 2. (24)

The bandwidth dimensioning problem consists of determination of link rates L1 andL2 that guarantee the service objective.

Proposition 3. If

L1 = κ[pM1 + qM2], (25)

L2 = κ[(1− p)M1 + (1− q)M2

], (26)


where κ > 1, and if the conditions of statement 1 in theorem 2 pertain, then theper-class throughput objective (24) is satisfied.

Note that for the per-class throughput objective (24), the dimensioned bandwidthonly depends on the given objective Mk and the routing via p and q, and not on anyother system parameters, other than via the conditions of statement 1 in theorem 2.

Proof. Substituting (19) in (2) and using (3) we have

ρki =Nλkfkeki

Li. (27)

Substituting (27) and (20) into (22) yields

θk ≡ Tkfk = Nλkfkβk. (28)

One of the conditions of statement 1 in theorem 2 is that ∆ ≡ ρ11ρ22 − ρ12ρ21 6= 0.From (27) and (20)

∆ =N2λ1λ2f1f2

L1L2(e11e22 − e12e21) =

N2λ1λ2f1f2

L1L2(p− q). (29)

Thus, we require p 6= q. Another condition of statement 1 in theorem 2 is that thesystem (13) has a positive solution βk. For ρki in (27), this condition implies

β1 =1

Nλ1f1· (1− q)L1 − qL2

p− q > 0, (30)

β2 =1

Nλ2f2· pL2 − (1− p)L1

p− q > 0. (31)

Substituting βk, k = 1, 2, from (30) and (31) into (28) yields

θ1 =(1− q)L1 − qL2

p− q , (32)

θ2 =pL2 − (1− p)L1

p− q , (33)

where θk > 0, k = 1, 2. Substituting (25) and (26) in (32) and (33) we get

θk = κMk, k = 1, 2,

and condition (24) is implied by the definition of θk in (28) since κ > 1.

5. Proofs

In this section we first establish in three lemmas properties of the maximum ofthe function F (x). Then we prove the two theorems using these properties.


5.1. Properties of the maximum

Lemma 4. If F (x) has a maximum at point x∗ with x∗ki > 0, k, i = 1, 2, then thesystem of linear equations (13) has such a solution βk ∈ (0,αk), k = 1, 2, thatthe system of linear equations (14) has a solution γi > 0, i = 1, 2. Moreover,x∗ki = ρkiβkγi.

Proof. Without the loss of generality we can assume that∑

i x∗ki < αk, k = 1, 2,

since ∂F (x)/∂xki → −∞ as∑

i xki → αk, k = 1, 2.Denote x = (x1,x2), y = (y1, y2) and consider the following auxiliary problem:

Maximize

H(x, y, x) =∑i

(xi lnxi − xi) +∑k,i


xki lnxki −∑k

yk ln yk (34)

subject to constraints∑k

xki = xi,∑i

xki + yk = αk, k, i = 1, 2, (35)

on the set

xki > 0, yk > 0, k, i = 1, 2. (36)

It is clear that x∗ provides a solution for the auxiliary optimization problem. On theother hand, the classical Lagrange multiplier method provides the following necessaryconditions for a local maximizer for the auxiliary problem [10, theorem 7.2.1]. Thereexist (γ1, γ2,β1,β2) such that

∇H(x∗, y∗, x∗

)+∑i

γ1i∇(∑

k

x∗ki − x∗i)

+∑k

β1k∇(∑

i

x∗ki + y∗k − αk)

= 0

or

lnx∗i − ln γi = 0, i = 1, 2,

− ln y∗k + lnβk = 0, k = 1, 2,

ln ρki − lnx∗ki + ln γi + lnβk = 0, i, k = 1, 2,

where

γi = exp(γ1i

), βk = exp

(β1k − 1

).

Thus, we have

γi = x∗i , βk = y∗k, x∗ki = ρkiγiβk, k = 1, 2. (37)

We have from (37) and the definition of xi that (β1,β2) satisfy the system (13).Furthermore, (37) and the condition

∑i x∗ki + y∗k = αk imply that (γ1, γ2) satisfy the

system (14). Now, to complete the proof we note that γi and βk satisfy the requiredconstraints by their definition in (37).


Lemma 5. Let x = (x1,x2), y = (y1, y2),

S =

x: x1 > 0, x2 > 0,

∑i

xi 6∑k

αk

and for any x ∈ S

D(x) =

(y, x):

∑k

xki = xi,∑i

xki + yk = αk, yk > 0, xki > 0, k, i = 1, 2

.

Let

G(y, x) =∑k,i


xki lnxki −∑k

yk ln yk,

K(x) = max(y,x)∈D(x)

G(y, x)

and

R(x) =∑i

(xi lnxi − xi) +K(x).

Then

maxx∈C

F (x) = maxx∈S

R(x). (38)

Moreover, let

xi > 0 and∑i

xi <∑k

αk, i = 1, 2. (39)

Then a maximum point (y0, x0) of G(y, x) in D(x) satisfies

y0i > 0, x0

ki > 0, k, i = 1, 2,

and∂R(x)∂xi

= ln∑k

ρkiy0k. (40)

Proof. G(y, x) is a strictly concave function on convex set D(x), and this set has aninterior point in the nondegenerate case when at least one of xi > 0. Therefore, forany fixed x ∈ S, G(y, x) has the unique maximum, and function K(x) is well defined.Now, (38) follows from the definition of R(x).

Under conditions (39) the function G cannot have a maximum in D(x) when oneof xki = 0 or yk = 0 since partial derivatives of G with respect to xki and yk tendto ∞ as the respective variable approaches 0. The Lagrangian for the maximizationproblem of G under constraints (35) on the set (36) is the function

L(y, x, γ1, γ2,β1,β2) = G(y, x)+

∑i

γi(∑

k

xki−xi)

+∑k

βk(∑

i

xki+yk−αk).


By the Kuhn–Tucker theorem there is a vector (γ10 , γ2

0 ,β10 ,β2

0 ) such that

G(y0, x0) = max

xki>0,yk>0L(y, x, γ1

0 , γ20 ,β1

0 ,β20

). (41)

The necessary conditions for a local maximum of L imply

− ln y0k + lnβ0

k = 0, k = 1, 2,

ln ρki − lnx0ki + ln γ0

i + lnβ0k = 0, i, k = 1, 2,

where

γ0i = exp

(γi0), β0

k = exp(βk0 − 1

).

Thus, we have

β0k = y0

k, x0ki = ρkiγ

0i β

0k, k = 1, 2. (42)

From (42) and the definition of xi we have

γi0 = lnxi − ln∑k

ρkiy0k.

Hence, (40) follows from the definition of R(x) and (41) since ∂L/∂xi = −γi.

Lemma 6. If ∆ 6= 0 and R(x) has a maximum on the axis x3−i = 0, then γ3−i < 0,i = 1, 2, where (γ1, γ2) is a solution of system (14).

Proof. First, we derive from lemma 5 that if (x1,x2) tends from inside S to a boundarypoint on axis x3−i = 0, and x∗i is a maximum point of R(x) at axis x3−i = 0, then

∂R(x)∂x3−i

∣∣∣∣xi=x∗i

→ ln∑k

ρk,3−iαkρk,iγ

∗i + 1

, (43)

where γ∗i is the unique positive solution of equation (17).Indeed, if x3−i → 0, then from the constraints in lemma 5 and (42) we have

y0k →

αk1 + ρkiγ

0i

, k = 1, 2. (44)

Substituting (44) in (40) we get

∂R(x)∂x3−i

→ ln∑k

ρk,3−iαkρk,iγ

0i + 1

. (45)

If xi = x∗i , then similarly

∂R(x)∂xi

∣∣∣∣xi=x∗i

→ ln∑k

ρk,iαkρk,iγ

0i + 1

= 0. (46)

Now (43) is implied by (45) and (46).


Next, let i = 1 and a maximum of R(x) be on axis x2 = 0. By (43) this implies∑k

ρk2αkρk1γ

∗1 + 1

< 1.

Then we show that system (14) has a solution with γ2 < 0. The case i = 2 is similarlyproved.

For t > 1 define a matrix ‖ρki(t)‖ which is obtained from ‖ρki‖ by multiplyingits second column by t. Let (β1(t),β2(t)) be a solution of the system∑

k

ρki(t)βk = 1, i = 1, 2,

and (γ1(t), γ2(t)) be a solution of the system∑i

ρki(t)γi =αk − βk(t)βk(t)

, k = 1, 2.

Let ‖σki‖ be the inverse matrix for ‖ρki‖. Then βk(t) = σ1k + σ2k/t and

tγ2(t) =∑k

σ2k

(αkβk(t)

− 1

)=∑k

(σ2k

αkσ1k + σ2k/t

− σ2k

).

Define

t0 =

[∑k

ρk2αkρk1γ

∗1 + 1

]−1

.

Note that 0 < βk(t) < αk for all t ∈ [1, t0] since βk(t) are monotone functions on[1, t0] while βk(1) and βk(t0) = αk/(ρk1γ

∗1 + 1) are in the interval (0,αk). Define

g(t) = tγ2(t). We have: g(1) = γ2,

g(t0) =∑k

σ2k

(αk

βk(t0)− 1

)=ρ11ρ21 − ρ21ρ11

∆γ∗1 = 0

and

g′(t) =∑k

αkσ22k

t2(σ1k + σ2k/t)2

is positive for t ∈ [1, t0]. Therefore, γ2 < 0.

5.2. Proof of theorem 1

First, we prove that the function F (x) has a maximum only on the set V =x ∈ C:

∑k,i xki = v∗, where v∗ is the unique root of equation (8) in the interval

(0,α1 + α2). If ρk1 = ρk2, then (40) implies that

∂R(x)∂x1

=∂R(x)∂x2

.


Therefore, R(x) = f (x1 + x2), where f (v) is a smooth function of one variable on[0,α1 +α2]. By condition ρ1α1 +ρ2α2 > 1 the function f (v) cannot have a maximumat v = 0 as f ′(0) > 0 by (40). The function f (v) cannot have also a maximum atv = α1 +α2 because ∂F (x)/∂xki → −∞ as x1 +x2 → α1 +α2. Therefore, f (v) hasa maximum at a point v∗ ∈ (0,α1 +α2). Consider a set of x∗ki > 0, k, i = 1, 2, whosesum is v∗. By lemma 4 x∗ki = ρkiγiβk, (β1,β2) satisfy (13) and (γ1, γ2) satisfy (14).Under the condition ρki ≡ ρk we have from (14)

βk =αk

1 + ρk(γ1 + γ2), k = 1, 2. (47)

By substituting (47) in the first equation of (13) and using the equation x∗i = γi, i =1, 2, we see that v∗ satisfies equation (8). To complete the proof note that equation (8)has a single solution in the interval (0,α1 + α2) as under conditions of the theoremthe left hand side of (8), denoted by h(v), has the following properties: h(0) > 1,h(α1 + α2) < 1 and h′(v) < 0 for v ∈ (0,α1 + α2).

Next, we prove (7), (9) and (10). Assuming r11 = r12 = r1, r21 = r22 = r2 andusing (1) we have

P

Q11 = n11,Q12 = m− n11,

∑k,i

Qki = l

=

1Ω

N1!(N1 −m)!

· N2!(N2 − (l −m))!

rl2∑

n1+n2=l

n1!n11!n21!

· n2!n12!n22!

ρm

=rl2Ω

N1!(N1 −m)!

· N2!(N2 − (l −m))!

(l + 1m+ 1

)ρm, (48)

where ρ = r1/r2. The last equality is obtained from the identity(l + 1m+ 1

)=

l−m∑i=0

(i+ j

i

)(l − j − il −m− i

), 0 6 j 6 m 6 l. (49)

Using the relation that (l + 1m+ 1

)=

(l + 1l −m

)and rewriting l −m as m, identity (49) can be rewritten as(

l + 1m

)=

m∑i=0

(i+ j

i

)(l − j − im− i

), 0 6 m+ j 6 l, 0 6 m, j.

Further, replacing l by l + j +m, we have(l + j +m+ 1

m

)=

m∑i=0

(i+ j

i

)(l +m− im− i

), m, l, j > 0.


This formula is nothing but (12.16) in [3, p. 65]. Apparently, the formula representsthe fact that the convolution of negative binomial distributions is binomial. From (48)we have

P

Q11 +Q12 = m,

∑k,i

Qki = l

= (m+ 1)rl2Ω· N1!

(N1 −m)!· N2!

(N2 − (l −m))!

(l + 1m+ 1

)ρm (50)

and

P

Q11 = j | Q11 +Q12 = m,

∑k,i

Qki = l

=

1m+ 1

, j = 0, 1, . . . ,m. (51)

Similar to [11], one can derive from (50) using assumptions (3) the following asymp-totic representation:

p(l,m) = P

Q11+Q12 = m,

∑k,i

Qki = l

= c(N ) exp

NΨ(v,u)+O(lnN )

, (52)

where v = l/N , u = m/N , c(N ) does not depend on (u, v) and

Ψ(v,u) = v ln ρ2 − v + v ln v + u ln ρ− (α1 − u) ln(α1 − u)− u lnu

− (v − u) ln(v − u)− (α2 − v + u) ln(α2 − v + u). (53)

We prove that the function Ψ(v,u) has a unique maximum inside

Γ =

(v,u): v ∈ (0,α1 + α2), u ∈ (0,α1), (v − u) ∈ (0,α2)

,

and the maximum point (v∗,u∗) is defined by the unique solution of equation (8)and (10). Indeed, the function Ψ(v,u) is strictly concave in Γ because its secondderivative

Ψvv = −(

1v − u −

1v

)− 1α2 − (v − u)

< 0, (v,u) ∈ Γ,

and the determinant of matrix of the second derivatives

|Ψvu| =v − u

uv(α2 − (v − u))+

u

v(α1 − u)(v − u)+

1(α1 − u)(α2 − (v − u))

> 0,

(v,u) ∈ Γ.

Hence, the function Ψ(v,u) has a single maximum inside Γ if the system of twoequations, defined by the necessary conditions for a local maximizer, has a solutionin Γ. The necessary conditions for a local maximizer of Ψ(v,u) have the followingform:

Ψ′v(v,u) = ln ρ2 + ln v − ln(v − u) + ln(α2 − (v − u)

)= 0,

Ψ′u(v,u) = ln ρ− lnu+ ln(α1 − u) + ln(v − u)− ln(α2 − (v − u)

)= 0,


orv − u

α2 − (v − u)= ρ2v, (54)

α1 − uu

· v − uα2 − (v − u)

=ρ2

ρ1. (55)

(54) and (55) can be rewritten as (8) and (10) as follows. Substitution of ρ2v insteadof the second fraction in the left hand side of (55) yields α1/u− 1 = 1/(ρ1v) or

u =α1ρ1v

1 + ρ1v= α1 −

α1

ρ1v + 1, (56)

which is (10). Using (56) to substitute out u in (54), and rearranging (54) yields (8).As shown above, (8) has a unique solution v∗ in (0,α1 + α2). Since v∗ is positive,then u∗ given by (56) is in (0,α1). Lastly, since the right hand side of (54) is positive,then (54) implies that v∗ − u∗ ∈ (0,α2). Thus, the pair (v∗,u∗) is the unique solutionto the first order conditions and is in Γ.

Finally, using the representation (52) and the fact that the function Ψ(v,u) has aunique maximum point (v∗,u∗) inside Γ, one can prove similar to [11] that(

1N

∑k,i

Qki,1N

∑i

Q1i

)D→(v∗, u∗

). (57)

Convergence in probability (7) and (9) is implied by convergence in distribution in (57)because the limit is deterministic. Now, (7), (9) and (51) imply the convergence (11)to the uniform distribution. (12) is similarly proved.

5.3. Proof of theorem 2

We prove below that under conditions of statements 1, 3 and 4 the function F (x)has a unique maximum point x∗ on C and identify its positive and zero components.Moreover, we prove that F (x) cannot have a maximum at a point, where xki = 0 whilexi > 0. Hence representation (4) implies the convergence Q/N P→ x∗, [11], whilethe positive and zero components of x∗ identify the bottleneck and non-bottleneck PSstations, respectively.

1. F (x) is a continuous function on a closed set C, and, therefore, it has amaximum on C. However, F (x) cannot have a maximum on the boundary x1 + x2 =α1 +α2 or at a point, where xki = 0 while xi > 0. The first statement is true because∂F (x)/∂xki → −∞ as x1 + x2 → α1 + α2. The second statement is true becauseby lemma 5 under conditions (39) the function G cannot have a maximum in D(x)when one of xki = 0. Moreover, by lemma 6 F (x) cannot have a maximum at theboundaries x1 = 0 or x2 = 0. Thus, F (x) has a maximum point x∗ inside C withx∗ki > 0. By lemma 4 the maximum point is unique and given by (15) as ∆ 6= 0.

2. If equations (13) have a solution βk ∈ (0,αk), then the condition ∆ = 0implies ρki ≡ ρk, and the results of the statement follow from theorem 1.


3. We consider cases (i) and (ii) separately.(i) Here we prove that the function F (x) has in C the unique maximum point

(x∗1i,x∗2i, 0, 0) defined by equations (18) and (17).

Since∑

k ρk2αk 6 1 the system of equation (13) does not have a positive solutionwith βk < αk. By lemma 4 F (x) cannot have a maximizing x∗, where x∗ki are allpositive. F (x) is a continuous function on a closed set C and, therefore, it has amaximum on C. However, by the same arguments as before F (x) cannot have amaximum on the boundary x1 + x2 = α1 + α2 or at a point, where xki = 0 whilexi > 0. Thus, F (x) has a maximum at the boundaries x1 = 0 or x2 = 0.

Similar to the proof of lemma 4 we consider the following auxiliary problem:Maximize

Hi(xi, y,x1i,x2i) = (xi lnxi−xi) +∑k

xki ln ρki−∑k

xki lnxki−∑k

yk ln yk (58)

subject to constraints∑k

xki = xi, xki + yk = αk, k = 1, 2,

on the set

xi > 0, y1 > 0, y2 > 0.

The necessary conditions for a local maximizer for the auxiliary problem give

γi = x∗i , βk = y∗k, x∗ki = ρkiγiβk, k = 1, 2. (59)

We have from (59) and the definition of xi that (β1,β2) satisfy the system (13).Furthermore, (59) and the condition x∗ki + y∗k = αk imply that

βk =αk

ρkiγi + 1, k = 1, 2. (60)

Substituting (60) in (13) we obtain equation (17), where the lower index in γ is omitted.Denote the left hand side of equation (17) by φi(γ). Function φi(γ) in monotonicallydecreasing on [0,α1 + α2] and φi(0) =

∑k ρkiαk. Therefore, if

∑k ρkiαk < 1, then

equation (17) does not have a solution. This means that the function F (x) may havea maximum only when xi = 0. If

∑k ρkiαk = 1, then equation (17) has the only

solution at xi = 0. This implies that the function F (x) may have a maximum onlywhen xi = 0. If

∑k ρkiαk > 1, then equation (17) has the unique solution γ∗i since

φ(α1 + α2) < 1. Substituting γi = γ∗i in (60) we obtain β∗k and finally (see (59))positive components of the maximum point x∗ki = γ∗i β

∗k that gives (18).

(ii) Here we prove that the function F (x) has the unique maximum at axisx3−i = 0 if condition (16) is satisfied. (18) is proved similarly to that in (i).

Since the conditions of statements 1 and 2 are not satisfied, the maximum of thefunction F (x) is at one of the axes x1 = 0 or x2 = 0. Condition

∑k ρkiαk > 1,

i = 1, 2, implies that the function F (x) cannot have a maximum at the origin. This


is because of relation (38), where both partial derivatives of the function R(x) givenby (40) are positive at x = 0. Moreover, one can prove as before that F (x) cannothave a maximum on the boundary x1 + x2 = α1 + α2 or at a point, where xki = 0while xi > 0.

First, we consider the case when system (13) does not have a solution βk ∈(0,αk). Define the two following sets:

Z =

z: 0 6 zk 6 αk,

∑k

ρk1zk = 1

,

W =

w: 0 6 wk 6 αk,

∑k

ρk2wk = 1

.

Our assumption implies that either

maxz∈Z

∑k

ρk2zk < 1 (61)

or

minz∈Z

∑k

ρk2zk > 1. (62)

It is easy to see that for i = 1, condition (16) is satisfied if and only if (61) holds. Fori = 2, condition (16) is satisfied if and only if (62) holds. This is because our assump-tion implies that maxw∈W

∑k ρk1wk < 1 if (62) holds, and minw∈W

∑k ρk1wk > 1

if (61) holds. Hence, under condition (16), the maximum of function R(x) can beonly on axis x3−i = 0 because this is the only case when the partial derivative (43) isnegative.

Next, we consider the case when ∆ 6= 0, and system (13) has a solution βk ∈(0,αk) but system (14) has a solution with γ1 6 0 or γ2 6 0. Hence, by lemmas 4and 5 the function R(x) does not have a maximum inside S. By lemma 6 the functionR(x) cannot have local maximums on both axes simultaneously. These two facts provethe statement.

4. It was proved in (i) that if∑

k ρkiαk < 1, then function F (x) may have amaximum only when xi = 0. This implies that x∗ = 0 is the unique maximum ofF (x) on the set C.

6. Generalizations

We covered the bottleneck analysis for all cases that can occur when K = M = 2.When M > 2 and K > 2 the bottleneck analysis becomes more complicated as manymore cases are possible and their complete exposition is beyond the scope of thispaper. However, in this section we state the results for an important subset of thecases, where bottleneck groups are excluded. Generalizations described in this sectionare based on comparison of the case of K = M = 2 with the previously studied cases


of K = 1, M > 1 and M = 1, K > 1 [7,9,11]. These generalizations can be provedby the same technique that is used in section 5 for K = M = 2.

In the case K = 1, M > 1, the network consists of one IS station and M singleservers (SS) numbered as 1, . . . ,M . SS 1 is a bottleneck if (see [7])

ρ1 > 1 and ρ1 > ρi, i = 2, . . . ,M. (63)

(Only the second index is used here since K = 1.) When there are multiple classes,the load at PS station i is defined as a linear combination of the per-class loads, andstation i could be a bottleneck or belong to a bottleneck group if its load exceeds 1:∑

k

ρkiαk > 1. (64)

In the case M = 1, K > 1, the network consists of one IS station with K classes andone PS station. The bottleneck condition (see [9,11]) is

∑k ρkαk > 1. (Only the first

index is used here since i = 1.) Under this condition, the equation (cf. (17))∑k

ρkαkρkγ + 1

= 1 (65)

has the unique positive solution γ∗, and x∗k is given by equation (18) with omittedindex i.

When both K,M > 1 the bottleneck identification becomes more complicatedas one can see from theorem 2. This is because more than one bottleneck may existand, in general, ordering of the loads for all PS stations (cf. (63)) does not identifythe bottleneck even in the case of one bottleneck (see statement 3 in theorem 2). Notethat (i) in statement 3 and statement 4 of theorem 2 apply for arbitrary K, M and arealready so stated. Moreover, statement 1 in theorem 2 directly generalizes for K > 2if K = M .

In general, one can solve (14) with respect to βk and substitute this solution in(13), which gives one system of nonlinear equations∑

k

ρkiαk1 +

∑j ρkjγj

= 1 (66)

with respect to γi instead of two systems of linear equations. Equations (66) are validonly for those i that correspond to bottleneck PS stations for which γi > 0. Fornon-bottleneck nodes γj = 0 in (66). To identify the bottleneck nodes it is necessaryto find the maximum subset BL = (i1, . . . , iL) of L (6 K) PS stations for which thecondition (64) is satisfied, and the system∑

k

ρkiαk1 +

∑j∈BL ρkjγj

= 1, i ∈ BL, (67)


has a positive solution γ∗i > 0, i ∈ BL such that∑k

ρkiαk1 +

∑j∈BL ρkjγ

∗j

< 1, i /∈ BL (68)

(cf. (16) and (17)). These two conditions can be used for bottleneck identificationfor any M and K, and, in particular, in the case when one of the systems of linearequations in the generalization of statement 1 in theorem 2 does not have a positivesolution, i.e., not all PS stations are bottlenecks. Moreover, for K > M they cannotbe simplified.

The maximum number of bottleneck stations and groups is min(M ,K). WhenM > K one can find K bottleneck nodes (if they exist) by solving linear equations (13)and (14) for different subsets BK = (i1, . . . , iK) of K nodes, i.e., i ∈ BK in (13)and (14). The subset B∗K is the bottleneck subset if both systems have positive solutionsfor i ∈ B∗K and, in addition, ∑

k

ρkiβ∗k < 1, i /∈ B∗K . (69)

For a single class network K = 1 and under conditions (63), B1 = (1) and β1 =1/ρ1, which in turn implies inequalities (69). However, we do not know whethercondition (69) can be further simplified for a general multiple class network.

Finally, for M > 2 more than one bottleneck group may exist and theorem 1can be generalized for each group. Thus, in general, a multiple class network withmultiple PS stations may have bottleneck nodes, bottleneck groups and non-bottlenecknodes.

Acknowledgements

We would like to thank Ward Whitt for his insightful discussions on the resultsof the paper, and Kathy Meier-Hellstern, Dave Houck, and Pat Wirth for their carefulreview of an earlier draft of the paper. We also thank the referee for helpful comments,which have improved the presentation of the results and for probabilistic interpretationof identity (49).

References

[1] A. Berger and Y. Kogan, Dimensioning bandwidth for elastic traffic high-speed data networks,submitted for publication.

[2] A. Birman and Y. Kogan, Asymptotic evaluation of closed queueing networks with many stations,Commun. Statist. Stochastic Models 8 (1992) 543–564.

[3] W. Feller, An Introduction to Probability Theory and Its Application, Vol. 1, 3rd ed., Revisedprinting (Wiley, New York, 1970).

[4] M. Freidlin and A. Wentzell, Random Perturbations of Dynamical Systems (Springer, New York,1984).


[5] F.P. Kelly, The dependence of sojourn times in closed queueing networks, in: Mathematical Com-puter Performance and Reliability, eds. G. Iazeolla, P.J. Courtois and A. Hordijk (North-Holland,Amsterdam, 1984) pp. 111–121.

[6] Y. Kogan, Another approach to asymptotic expansions for large closed queueing networks, Oper.Res. Letters 11 (1992) 317–321.

[7] Y. Kogan and A. Birman, Asymptotic analysis of closed queueing networks with bottlenecks, in:Performance of Distributed Systems and Integrated Communication Networks, eds. T. Hasegawa,H. Takagi and Y. Takahashi, IFIP Transactions C-5 (North-Holland, Amsterdam, 1992) pp. 265–280.

[8] Y. Kogan and A. Yakovlev, Asymptotic analysis for closed multichain queueing networks withbottlenecks, Queueing Systems 23 (1996) 235–258.

[9] J. McKenna and D. Mitra, Integral representation and asymptotic expansions for closed Markovianqueueing networks: Normal usage, Bell Syst. Tech. J. 61 (1982) 661–683.

[10] A.L. Peressini, F.E. Sullivan and J.J. Uhl Jr., The Mathematics of Nonlinear Programming (Springer,New York, 1988).

[11] B. Pittel, Closed exponential networks of queues with saturation: The Jackson-type stationarydistribution and its asymptotic analysis, Math. Oper. Res. 6 (1979) 357–378.

[12] Private network–network interface specification, Version 1.0, ATM Forum (March 1996).

Bottleneck analysis in multiclass closed queueing … · Queueing Systems 31 (1999) 217–237 217 Bottleneck analysis in multiclass closed queueing networks and its application Arthur

Documents